32 Painlevé Transcendents Applications 32.13 Reductions of Partial Differential Equations 32.15 Orthogonal Polynomials

§32.14 Combinatorics

ⓘ

Keywords:: Gaussian unitary ensemble, Hermitian matrices, Painlevé transcendents, applications, combinatorics, distribution function, limiting distribution of eigenvalues, random matrix theory
Notes:: See Baik et al. (1999).
Permalink:: http://dlmf.nist.gov/32.14
See also:: Annotations for Ch.32

Let $S_{N}$ be the group of permutations $\boldsymbol{\pi}$ of the numbers $1,2,\dots,N$ (§26.2). With $1\leq m_{1}<\cdots<m_{n}\leq N$ , $\boldsymbol{\pi}(m_{1}),\boldsymbol{\pi}(m_{2}),\dots,\boldsymbol{\pi}(m_{n})$ is said to be an increasing subsequence of $\boldsymbol{\pi}$ of length $n$ when $\boldsymbol{\pi}(m_{1})<\boldsymbol{\pi}(m_{2})<\cdots<\boldsymbol{\pi}(m_{n})$ . Let $\ell_{N}(\boldsymbol{\pi})$ be the length of the longest increasing subsequence of $\boldsymbol{\pi}$ . Then

32.14.1		$\lim_{N\to\infty}\mathrm{Prob}\left(\frac{\ell_{N}(\boldsymbol{\pi})-2\sqrt{N}% }{N^{1/6}}\leq s\right)=F(s),$
ⓘ Symbols: $\boldsymbol{\pi}(m)$ : permutations, $\ell_{N}(\boldsymbol{\pi})$ : length and $F(s)$ : distribution function Permalink: http://dlmf.nist.gov/32.14.E1 Encodings: TeX, pMML, png See also: Annotations for §32.14 and Ch.32

where the distribution function $F(s)$ is defined here by

32.14.2		$F(s)=\exp\left(-\int_{s}^{\infty}(x-s)w^{2}(x)\,\mathrm{d}x\right),$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $x$ : real and $F(s)$ : distribution function Referenced by: §32.14 Permalink: http://dlmf.nist.gov/32.14.E2 Encodings: TeX, pMML, png See also: Annotations for §32.14 and Ch.32

and $w(x)$ satisfies $\mbox{P}_{\mbox{\scriptsize II}}$ with $\alpha=0$ and boundary conditions

32.14.3	$\displaystyle w(x)$	$\displaystyle\sim\operatorname{Ai}\left(x\right),$
$x\to \infty$ ,
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\sim$ : asymptotic equality and $x$ : real Permalink: http://dlmf.nist.gov/32.14.E3 Encodings: TeX, pMML, png See also: Annotations for §32.14 and Ch.32
32.14.4	$\displaystyle w(x)$	$\displaystyle\sim\sqrt{-\tfrac{1}{2}x},$
$x\to-\infty$ ,
ⓘ Symbols: $\sim$ : asymptotic equality and $x$ : real Permalink: http://dlmf.nist.gov/32.14.E4 Encodings: TeX, pMML, png See also: Annotations for §32.14 and Ch.32

where $\operatorname{Ai}$ denotes the Airy function (§9.2).

The distribution function $F(s)$ given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of $n\times n$ Hermitian matrices; see Tracy and Widom (1994).

See Forrester and Witte (2001, 2002) for other instances of Painlevé equations in random matrix theory.

32.13 Reductions of Partial Differential Equations 32.15 Orthogonal Polynomials

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov